---
tag: 2DSKCVDFAS
kind: definition
name: Montel space
msc: [46A11]
defines: NWMYW2WK9S
uses: [ZE6DXEJB8H, 3BAJ8ZTVPX, 0ZCK8DQYN1, 0R6EWFKRCF]
sources:
  - url: https://en.wikipedia.org/wiki/Montel_space
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a Hausdorff [[barrelled space|ZE6DXEJB8H]] over $\mathbb{K}$.
$E$ is a *Montel space* if every [[closed|0ZCK8DQYN1]] and [[bounded|3BAJ8ZTVPX]] subset of $E$ is [[compact|0R6EWFKRCF]].

## Remarks

A locally convex space in which closed bounded sets are compact, without the barrelled condition, is called *semi-Montel*.
Example: for an open set $\Omega \subseteq \C$, the space $H(\Omega)$ of holomorphic functions with the seminorms $p_K(f) = \sup_{z \in K} \abs{f(z)}$, $K \subseteq \Omega$ compact, is a Montel space.
It is a Fréchet space, hence barrelled.
A bounded set is a family that is uniformly bounded on each compact set, so [[Montel's theorem|ZVFVBVJMCB]] gives the compactness of closed bounded sets.
Non-example: an infinite-dimensional Banach space is barrelled, but its closed unit ball is closed and bounded and not compact, by Riesz's lemma.
Every Montel space is reflexive, and its strong dual is again a Montel space.